1. Field of the Invention
This invention relates generally to sound amplification systems, and more particularly to a stable filter for eliminating acoustic feedback in such systems.
2. Description of the Prior Art
The present invention finds applications in a variety of sound amplification systems, including public address systems and other situations in which there is substantial sound amplification and related feedback oscillation problems. However, because the present invention is especially well suited to hearing aid applications, the invention will be explained in the context of a hearing aid system.
As is well known in the hearing aid art, each instrument has associated with it the capability of oscillation when some of the output signal from the instrument is fed back acoustically into the microphone. This situation is illustrated in block diagram form in FIG. 1, wherein numeral 14 designates a hearing aid with a transfer function H.sub.HA (f) having an input x on line 10 and an output y on line 15. For purposes of explanation, it will be assumed that this particular hearing aid system is a linear system. A feedback path with a transfer function H.sub.FB (f) is generally obtained by either mechanical coupling of the receiver and the microphone transducers in the hearing aid itself, or more commonly, by virtue of a leak between the ear canal and the ear mold. For mechanical coupling, the feedback transfer function can be modeled as H.sub.FB (f)=.alpha.. For ear canal leakage, the feedback transfer function can be modeled as H.sub.FB (f)=.alpha.e.sup.-j2.pi.f.tau.. Both of the above-noted models are based on the assumption that .vertline..alpha..vertline..ltoreq.1. This feedback transfer function is shown in FIG. 1 by the numeral 16 which takes an input from the output line 15 and applies its output signal as one input to an adder 12 which has the input line 10 as its other input. The transfer function of the total system shown in FIG. 1 thus becomes: ##EQU2##
In order to obtain stability for the above feedback system, the poles of the above-noted transfer function, equation 1, are required to have negative real parts. If the poles of this transfer function do not have negative real parts, then instability will occur at the characteristic roots of the complex denominator polynomial: EQU 1-H.sub.FB (f).multidot.H.sub.HA (f)=0 (2)
This equation translates to two real counterparts: EQU 1-.vertline.H.sub.FB (f).vertline..multidot..vertline.H.sub.HA (f).vertline.=0 (3)
and EQU .theta..sub.FB (f)+.theta..sub.HA (f)=n.multidot.2.pi.; in radians; n=0, 1, 2 (4)
Equations 3 and 4 determine when and where this feedback instability will occur. In hearing aids which are designed for providing gain, the quantity .vertline.H.sub.HA (f).vertline. may be very large. For both feedback path models noted above, the feedback magnitude function .vertline.H.sub.FB (f).vertline.=.alpha.. Thus, according to equation 3, .alpha. must be made as small as possible in order to allow the highest hearing aid gain to be realized before feedback occurs. However, there are physical limits to the smallness of .alpha. because ear canal leaks cannot be totally prevented. Thus, according to equation 3, the magnitude function .vertline.H.sub.HA (f).vertline. representing the hearing aid gain must be limited if no feedback oscillation is to occur.
The frequency response of the total system with feedback is shown in FIG. 2 wherein there is a single oscillation feedback frequency f.sub.fb in the frequency band of interest. For a system with the frequency response shown in FIG. 2, it is apparent that a solution to this feedback oscillation problem would be to use a notch filter to modify the magnitude function .vertline.H.sub.HA (f).vertline., such that the resulting transfer function vanishes at the oscillation frequency. This notch filter may be represented by a transfer function: EQU H.sub.NOTCH (f)=.vertline.H.sub.NOTCH (f).vertline.e.sup.j.theta.NOTCH (f) (5)
The modified block diagram now appears as shown in FIG. 3. The only change in this figure has been the addition of the transfer function 18 representing the transfer function of the notch. Using a derivation analogous to the derivation for equations 3 and 4, the following stabiity equations can be obtained for the system of FIG. 3: EQU 1-.vertline.H.sub.FB (f).multidot..vertline.H.sub.HA (f).vertline..multidot..vertline.H.sub.NOTCH (f).uparw.=0 (6)
and EQU .theta..sub.FB (f)+.theta..sub.HA (f)+.theta..sub.NOTCH (f)=n.multidot.2.pi.n=0, 1, 2, (7)
By making the magnitude function .vertline.H.sub.NOTCH (f).vertline. very small for a particular oscillation frequency it appears possible to eliminate that oscillation frequency.
We have not discussed the equations 4 and 7 in connection with the determination of the proper notch transfer function. However, these equations determine where in the frequency range the oscillations will occur, if they occur at all.
The conventional notch filter is typically an analog filter comprising a cascade of several second order sections each with a transfer function as follows: ##EQU3## Each of these second order functions will exhibit a transfer function magnitude .vertline.H.sub.CONV (f).vertline. as shown in FIG. 4, and a phase characteristic as shown in FIG. 5. It is clear from a review of FIG. 5, that at the notch frequency .omega..sub.N, where .omega..sub.N /.omega..sub.0 =1, the phase characteristic jumps by .pi. radians. Thus, exactly at the notch frequency f.sub.N, where ideally the magnitude of the transfer function .vertline.H.sub.NOTCH (f).vertline. vanishes, the phase characteristic .theta..sub.CONV (f) is not precisely determined. The effect of the indeterminate nature of the phase characteristic of this notch filter is that the equation 7 will no longer be a limit on the number of frequency solutions to the equation 6 which will cause feedback oscillation. Since the high gain of the typical hearing aid will cause the equation 6 to be satisfied by a number of frequencies, the system will oscillate at any one of these frequencies and still satisfy equation 7.
In practical circumstances, the conventional notch filter can be slightly misadjusted, so as to have the notch occur a little above or a little below the initial frequency of oscillation. A review of the phase characteristics shown in FIG. 5 demonstrates that the phase at the notch frequency now is determined. Therefore, the solutions to equations 6 and 7 will permit only a finite set of oscillation frequencies to occur. However, from a review of the magnitude of the transfer function shown in FIG. 4, it can be seen that this slight misadjustment will cause only a small gain reduction at the oscillation frequency f.sub.N. Thus, the corresponding gain increase in .vertline.H.sub.HA (f).vertline. will be limited to a few db at most.
Various attempts have been made to solve this acoustic feedback problem in conjunction with other design goals. By way of example, a copending application by Graupe, Beex and Causey, Ser. No. 660513, filed on Feb. 23, 1976, deals with an auto-regressive type or recursive filter for tailoring the frequency response of a filter to a desired frequency response required to compensate for the defects in the frequency spectrum of a particular listener. Any digital filter of the recursive type must have a method of sampling inputs (i.e., sound levels at various times) and combining the information gained from that sampling method to arrive at the control exercised by the system. Such digital filters of the recursive type not only sample and use data from the environment being sampled, but also make a computation based on prior computations. The filter disclosed in the above-noted application is characterized in that generally less samples are required in order to obtain the particular filter frequency response desired. Such a smaller number of required samples is made possible by the use of the recursive feedback of the system which provides additional terms which vary in accordance with Y.sub.(K-1) , Y.sub.(K-2), etc. This type of filter can be specifically designed to have a notch to remove acoustic feedback frequencies. However, this type of filter can be unstable at certain frequencies because of the poles in its transfer function. This instability would become apparent when the sampled inputs to the recursive filter are consecutively zero for several readings, because there will then be an output even though the inputs have remained zero over a prolonged period of time. In comparison with a moving-average filter this type of filter is generally more sensitive to parameter variations due to the fewer parameters used in the system. That is, the changes in frequency response caused by temperature drift and other component tolerances will be much larger in comparison, and in some cases, could drive a stable system to instability. Moreover, with a recursive filter as disclosed in the above-noted application it is not possible to obtain exact linear phase response. (See the Rabiner and Gold reference noted hereinafter, at page 206). Additionally, the mere existence of poles in a filter transfer function will cause that transfer function to have rapidly changing phase characteristics at the location of the poles. See Rabiner and Gold, page 824. Since a notch implementation in a recursive filter will generally require a pole location directly adjacent to the notch itself, this will cause the phase characteristic at the notch frequency to be subject to rapid changes, or even indeterminate behavior as in conventional analog filters.